Evolution is fundamentally a spatial process, as populations interbreed locally and adapt to their environments. This has profound implications for how we understand patterns of genetic diversity within populations, especially as we begin to obtain large amounts of genetic data from populations. Good models and predictions of these processes will be necessary to understand the pat- terns we see, for instance, by allowing us to identify loci underlying adaptive traits, and to distinguish local adaptation from other processes. I will create and study geographically explicit models of selection, incorporating parallel selective sweeps (soft sweeps), the hitchhiking of neutral variation along with a selected allele, and the newly-described phenomenon of allele surfing; and will analyze their expected signatures in population genomics datasets. This will begin to fill a major gap in existing population genetics theory, which has relatively few explicit spatial models, and even fewer that incorporate selection. Such tools will not only allow us to better reconstruct the history of adaptation in geographically distributed species such as our own, but will also help us to more precisely identify genomic loci responding to selection pressures, and so find loci involved in disease response and in physiological adaptation to local conditions. I will do this by building on existing theory from population genetics, probability, ecology, and statistical physics, in close conversation with the goals and problems of empirical researchers.